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Tentative Lecture Plan
 From the outset this is the plan of 2017.
 There will be some changes. I will update the plan as I lecture.
Week  Topic  Chapter  Pages  Remarks 

2  Introducution The basics: Metric and Banach spaces, duals, \(\ell^p\), HahnBanach and consequences  1 2  p. 58, Theorem A.1 (appendix), Brezis chp 1.1: Corollay 1.21.4  Introduction meeting. 1 lecture. 
3  Conclusion to HahnBanach, separable and reflexive spaces, Compactness in metric spaces, weak convergence, relation to strong convergence, BanachSteinhaus without proof  p. 710, 1516  
4  Weak compactnes, EberleinSmuljan's thm, weak * convergence, Helley's thm, Alaoglu's thm, strong compactness for functions: ArzelaAscoli's thm   p 1117  
5  Distribution theory Definitions, properties, operations, regular/singular operations(cont.), derivatives of regular distr., the fundamental theorem, convolution  3   Remark: We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other. 
6  Convolutions (cont.), convergence, approximations Primitive in 1D, equations in \(D'\), fundamental solutions  
7  Lebesgue Spaces Strong and weak \(L^p\), properties, inequalities. Convolutions, approximation in \(L^p\). Approximation in \(L^p\), compactness in \(L^p\)  4  4.1, Prop 4.4 and Thm 4.6, 4.2, 4.7, 4.3 p 8689  Proof of Kolmogorov: We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in HoldenRisebro: Front Tracking for Hyperbolic Conservation Laws. The classical proof is an approximation argument that reduces the proof to an application of ArzelaAscoli. OBS: 3 lectures this week, only one next week. 
8  Modes of convergence  p 5963  Only one lecture this week, three last week. 

9  Convergence and compactness in \(L^p,\ p\in(1,\infty)\). Convergence and compactness in\(L^1\), DunfordPettis with proof, uniform integrability, de la ValleePoussin. Convergence and compactness in \(L^\infty\) Examples.  p 6065 p 6575  My lecture notes on DunfordPettis and equiintegrability PDF Note that the proof of DunfordPettis in the notes of Holden lack the conclusion. The discussion on equiintegrability can not be found in Holden, see my notes. 

10  Radon measures, the space \(\mathcal M\), Weak * compactness in \(\mathcal M\) and the subspace \(L^1\) Sobolev Spaces Definitions, smooth approximations  5  p 7579 9597  This material is partly taken from Folland: Real Analysis chp 7, partly from Holden. Note: Riesz representation theorem, the space \(M\) needs be defined as the space of finite Radon measures. My lecture notes on Radon measure and compactness: PDF. 
11  Smooth approximations (cont.) straightening the boundary, extensions Extensions, restrictions/trace  9797, 179 (App B.3) 9799  OBS: Global approx up to boundary  the proof in Holden using straightening is not optimal and only gives \(C^1\) approximate functions. In the lectures I used the proof from Evans: PDEs chapter 5 that avoids straightening and give \(C^\infty\) approximations. My lecture notes on smooth approximation up the boundary and straightening the boundary: PDF. 

12  Restrictions/trace (cont), Sobolev inequalities, GagliardoNirenbergSobolev Sobolev inequalities: Gagliardo, Poincare. H\"older spaces, Morrey's inequality.  98102 102104 105107, 111112 110111, 118.  Obs: 3 lectures this week.  
13  General Sobolev inequalities, embedding, compactness in \(W^{1,p}\)  116118, 1819, 112114  Obs: 1 only one lecture this week. 

14  Compactness (cont.), Sobolev chain rule, finite differences.  114116, 126, 128  RellichKondrachov: In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces \(C^{0\,\gamma}\). The proof of the first part of RellichKondarchov's compactness theorem follows from extension, KolmogorovRiesz compactness theorem, and interpolation in \(L^p\). This way avoids the long regularization + ArzelaAscoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + ArzelaAscoli argument is exactly the argument we used in class to proof KolmogorovRiesz in the first place (but the proof in the notes of Holden is slightly different). My lecture notes from this week: General Sobolev inequalities and Strong comactness in \(W^{1,p}\) Strong comactness, chain rule, and difference quotients 

15  Application: To be decided 